State Standards: GLEs/GSEs National Content Standards:
Grade Span Expectations [Math]:
M(N&O)–10–8 Applies properties of numbers to solve problems, to simplify computations, or to compare and contrast the properties of numbers and number systems. M(F&A)–10–1 Identifies, extends, and generalizes a variety of patterns (linear and nonlinear) represented by models, tables, sequences, or graphs to solve problems. M(PRP)–HS–1 Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content and be able to:
• Expand the repertoire of problem-solving strategies and use those strategies in more sophisticated ways.
• Use technology whenever appropriate to solve real-world problems (e.g., personal finance, wages, banking and credit, home improvement problems, measurement, taxes, business situations, purchasing, and transportation).
• Formulate and redefine problem situations as needed to arrive at appropriate conclusions.
Grade Span Expectations [Physics]:
PS3 (7-8) 8: Students demonstrate and understanding of motion by… (8a)… measuring the distance and time for a given moving object and using those values to calculate speed. (8b)… Solving for any unknown in the expression v=d/t given values for the other two variables. (8c)… Differentiating between speed, velocity and acceleration.
PS3 (9-11) 8: Students demonstrate and understanding of forces and motion by… (8a)… Predicting and/or graphing the path of an object in different reference planes and explain how and why it occurs. (8b)… Using modeling, illustrating, and graphing to explain how distance and velocity change over time for a free falling object.
PS3 (Ext) 8: Students demonstrate an understanding of forces and motion by… (8aa)… Using a quantitative representation of how distance and velocity change over time for a free falling object. (8bb)… Using a quantitative representation of the path of an object that has both horizontal and free fall motion. (8cc)… Modeling, illustrating, graphing, and or quantitatively explaining the path of an object that has both horizontal and free fall motion.
Context of the Lesson Where does this lesson fit in the curriculum and instructional context? Is it the opening of a unit or a series of lessons?
This lesson will take place in the middle of a projectile motion unit. Students should have already been exposed to 1 dimensional kinematics problems and they should be proficient with solving problems involving objects moving in either the x or the y direction. They should also know how to break vectors into components using basic trigonometric ratios.
In the lesson preceding this lesson students will have been introduced to the “bullet shot/ dropped” theory. They will have experimentally discovered that the motion of an object in the x and y directions are entirely independent of one another. This idea is central to the study of projectile motion and teachers should not proceed in this unit if this concept is not fully understood by the students.
Finally, students should be able to solve mathematical equations involving 1 variable and they should be familiar with solving systems of equations involving 2 unknowns.
This lesson will provide students with the theoretical background that they will need in order to solve problems relating to projectile motion. It is certainly an essential direct instruction lesson that should precede any student investigations involving projectiles (range, time of flight, etc).
Opportunities to Learn
Differentiation: Materials, Learners and Environments
Plans to differentiate instruction:
In order to differentiate instruction during this particular lesson I have divided the instruction period into three segments. The first segment involves student presentations and class discussions. The second segment involves direct instruction from the teacher and guided practice problems for the students to complete individually. The final segment involves the students working in groups to solve several example problems with the teacher’s assistance. Visual representations of the concepts are also incorporated within the lesson. The three segments of the lesson are designed to be beneficial for all types of learners. Traditional learners will find the middle section of the lesson particularly beneficial while students that enjoy working cooperatively will find the final part of the lesson beneficial. I also continuously change the activities for the students to keep them engaged and to not allow them to get lost in their own worlds! In order to differentiate instruction for the more advanced students I have found several more advanced projectile motion problems (taken from an AP book) that are in the AP format. I will allow the students to tackle these problems once they finish the assignment in their groups. For the students that may be struggling I have developed a graphic organizer that they can use to help them follow the problem solving steps in order to successfully solve each of the assigned problems. I will also help them by providing them with a visual representation of the problems and by helping them identify what they know and what they are trying to find out. Accommodations and modifications: Physics 1 classes will be comprised of learners of all different levels and learners who prefer to learn in a variety of different ways. In order to accommodate for all of these different types of learners I will group the students based on their learning ability. I will group the advanced students together and group the students that may take longer to learn certain tasks together. I will keep in mind that these groups will change depending on what the task at hand is (some students are stronger with certain concepts than with others). I will also try to put different types of learners together in the same groups. For example if I am making a group of advanced students I will try to place a kinesthetic learner with a mathematical learner so that they can start to see the concepts from a different perspective. Environment factors: During the first activity the students will be seated at their desks and the groups will come to the front of the classroom to present their findings. During the second segment of the lesson they will remain at their desks while we take some notes and do some examples together as a class. Finally, during the final segment of the lesson the students will work in groups at the lab benches that are located around the perimeter of the classroom. Materials: ü Notes (attached) ü White boards
Objectives
Students will demonstrate their understanding of projectile motion by successfully completing several example problems together in groups.
Instructional Procedures
LAUNCH A: The first part of this lesson is designed to finalize the activity that was performed the previous day relating to the “bullet shot/ dropped theory”. Each group will have prepared a short presentation and will have a diagram of their experimental set up and a table containing their results on the whiteboard. After each group presents their results I will ask them several probing questions to gauge how well they understand the scientific process as well as the concepts being studied. “What were some sources of error that you think may have affected your results?” “How could your procedure be adapted if we were working with faster projectiles?” “What happened if you didn’t shoot the projectile perfectly horizontal?” “Are these results something that you would have expected?” After all of the groups have presented their findings to the class I will summarize the findings and reiterate the “bullet shot/dropped” theory. I will explain the implications of this idea and how it can be used to help us study projectiles. This theory implies that the x and the y directions are entirely independent of one another and can be mathematically solved separately. The only variable that remains the same for the two directions is the independent variable of time. I will instruct the students to take a seat and I will have them all open their notebooks so that these concepts can be formally presented to them. The notes that I will give the students can be seen on the pages that follow this lesson plan. I will begin by explaining what a projectile is and what “projectile motion” actually means. I will then review the equations that they used in 1 dimensional kinematics and will explain how these can be used for the x and y directions separately. “Why do we have two sets of equations for the x and y directions?” We will then discuss how these equations can be simplified for an object that is launched into the air. “What is the acceleration in the y direction for an object that is launched into the air?” “What is the acceleration in the x direction for an object that is launched into the air? Why? What are the implications of this?” I will then show them the simplified versions of the equations and we will proceed to review breaking components into vectors and finally we will do two examples together as a class. Prior to the examples however, I will distribute the graphic organizer that is attached and we will discuss/review the steps to problem solving because this is a critical skill that they must all master. “Does anyone remember the steps to problem solving? I have made a graphic organizer to help you remember them!” EXPLORE A: After distributing the graphic organizer and working through the two example problems together as a class, I will ask the students if they have any specific questions regarding the topics that we covered. I will then have the students work in groups to complete several problems on their own. I will inform them that I will be collected them at the end of the class and they will receive a completion grade for their work. I will require that they explicitly write out all of the steps that they used during the process. By this I mean that I will be checking to ensure that they explicitly identify what variable/variables are given in the problem and draw a diagram to go along with the given scenario. I will also be checking to make sure that they identify the appropriate equations and keep the variables in the equations until the very end (i.e make symbolic algebraic manipulations before plugging in numbers). Note: Each student will have to turn in their own work. If groups are having trouble with the task I will offer some guidance: “What are we trying to solve for in this problem?” “What would the diagram look like to go along with this situation?” “How can we figure out which equation will help us in this situation?” “Given the initial velocity and the angle that the projectile is launched at, what two variables can we determine using simple trigonometric ratios?” I will essentially guide the groups in the right direction with the problems but I will not explicitly do them for them. I will help them follow the problem solving steps and try to get them in the habit of always resorting back to those basic steps. If certain groups are finding the problems very simple, I will have more challenging problems to present to them. The problems that I will give to the students can be seen in the attached document and were found in “Barrons Guide to AP Physics” SUMMARIZE/SHARE A: Once all of the groups have finished “exploring” the sample problems we will reconvene as a class and I will ask them if they have any specific questions about the problems. If they do then we will address those and possibly do the problems together as a class. If there are no questions and there is still time remaining I will give the students a HW assignment from their textbook “Holt Physics” I will also explain to the students that they will have a short quiz at the beginning of the following period on basic projectile motion problems. HW: Pg 97 #1-4
Assessment
In order to assess my students during this investigation I will determine whether each student has successfully completed each of the objectives described above. Launch A: During the first portion of this lesson I will be able to observe the students as they present their findings from the investigation that was performed on the previous day. The students will be forced to vocalize their understanding of what they did and what they were able to show. In the process of doing this I will be informally assessing them on the presentation skills and on their knowledge of the content of the lesson. I will specifically be looked to see if they truly grasp the “bullet shot/ dropped” phenomenon. In this process I will also be able to assess whether or not the students work well in groups. I will take notes during each presentation and will determine if it will be necessary to spend more class time on the concepts that were taught the previous day. If I deem it necessary I will postpone the next instructional period and will elaborate on the previous concepts. It is very important to build a conceptual foundation before introducing mathematics into the physics classroom and I will use informal assessment to make sure that this happens. Explore/ Summarize A: During the group work I will walk around the room and assist the students. In the process of doing this I will be informally assessing the students on their understanding of the material. If some students seem to be struggling I will be sure to spend more time with their groups and attempt to clear up any misconceptions that they may have and fill in any gaps in their knowledge that may exist. I will also collect the HW and will formally assess each students ability to follow the problem solving process. I will not assess whether or not they got the problem correct (although I will check this as well) but rather I will make sure that they followed the correct procedure and identified the steps properly. I want to stress problem solving and critical thinking skills and I will constantly make sure that the students are showing all of their work and are breaking down problems effectively.
Reflections This section to be completed only if lesson plan is implemented.
Lesson Implementation: Was not able to implement this lesson yet.
Lesson #4: Projectile Motion Introduction
Lesson Title
Projectile Motion Introduction
National Content Standards:
M(N&O)–10–8 Applies properties of numbers to solve problems, to simplify computations, or to compare and contrast the properties of numbers and number systems.
M(F&A)–10–1 Identifies, extends, and generalizes a variety of patterns (linear and nonlinear) represented by models, tables, sequences, or graphs to solve problems.
M(PRP)–HS–1 Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content and be able to:
• Expand the repertoire of problem-solving strategies and use those strategies in more sophisticated ways.
• Use technology whenever appropriate to solve real-world problems (e.g., personal finance, wages, banking and credit, home improvement problems, measurement, taxes, business situations, purchasing, and transportation).
• Formulate and redefine problem situations as needed to arrive at appropriate conclusions.
Grade Span Expectations [Physics]:
PS3 (7-8) 8: Students demonstrate and understanding of motion by…
(8a)… measuring the distance and time for a given moving object and using those values to calculate speed.
(8b)… Solving for any unknown in the expression v=d/t given values for the other two variables.
(8c)… Differentiating between speed, velocity and acceleration.
PS3 (9-11) 8: Students demonstrate and understanding of forces and motion by…
(8a)… Predicting and/or graphing the path of an object in different reference planes and explain how and why it occurs.
(8b)… Using modeling, illustrating, and graphing to explain how distance and velocity change over time for a free falling object.
PS3 (Ext) 8: Students demonstrate an understanding of forces and motion by…
(8aa)… Using a quantitative representation of how distance and velocity change over time for a free falling object.
(8bb)… Using a quantitative representation of the path of an object that has both horizontal and free fall motion.
(8cc)… Modeling, illustrating, graphing, and or quantitatively explaining the path of an object that has both horizontal and free fall motion.
Where does this lesson fit in the curriculum and instructional context? Is it the opening of a unit or a series of lessons?
In the lesson preceding this lesson students will have been introduced to the “bullet shot/ dropped” theory. They will have experimentally discovered that the motion of an object in the x and y directions are entirely independent of one another. This idea is central to the study of projectile motion and teachers should not proceed in this unit if this concept is not fully understood by the students.
Finally, students should be able to solve mathematical equations involving 1 variable and they should be familiar with solving systems of equations involving 2 unknowns.
This lesson will provide students with the theoretical background that they will need in order to solve problems relating to projectile motion. It is certainly an essential direct instruction lesson that should precede any student investigations involving projectiles (range, time of flight, etc).
Differentiation: Materials, Learners and Environments
In order to differentiate instruction during this particular lesson I have divided the instruction period into three segments. The first segment involves student presentations and class discussions. The second segment involves direct instruction from the teacher and guided practice problems for the students to complete individually. The final segment involves the students working in groups to solve several example problems with the teacher’s assistance. Visual representations of the concepts are also incorporated within the lesson.
The three segments of the lesson are designed to be beneficial for all types of learners. Traditional learners will find the middle section of the lesson particularly beneficial while students that enjoy working cooperatively will find the final part of the lesson beneficial. I also continuously change the activities for the students to keep them engaged and to not allow them to get lost in their own worlds!
In order to differentiate instruction for the more advanced students I have found several more advanced projectile motion problems (taken from an AP book) that are in the AP format. I will allow the students to tackle these problems once they finish the assignment in their groups.
For the students that may be struggling I have developed a graphic organizer that they can use to help them follow the problem solving steps in order to successfully solve each of the assigned problems. I will also help them by providing them with a visual representation of the problems and by helping them identify what they know and what they are trying to find out.
Accommodations and modifications:
Physics 1 classes will be comprised of learners of all different levels and learners who prefer to learn in a variety of different ways. In order to accommodate for all of these different types of learners I will group the students based on their learning ability. I will group the advanced students together and group the students that may take longer to learn certain tasks together. I will keep in mind that these groups will change depending on what the task at hand is (some students are stronger with certain concepts than with others). I will also try to put different types of learners together in the same groups. For example if I am making a group of advanced students I will try to place a kinesthetic learner with a mathematical learner so that they can start to see the concepts from a different perspective.
Environment factors:
During the first activity the students will be seated at their desks and the groups will come to the front of the classroom to present their findings. During the second segment of the lesson they will remain at their desks while we take some notes and do some examples together as a class. Finally, during the final segment of the lesson the students will work in groups at the lab benches that are located around the perimeter of the classroom.
Materials:
ü Notes (attached)
ü White boards
Instructional Procedures
The first part of this lesson is designed to finalize the activity that was performed the previous day relating to the “bullet shot/ dropped theory”. Each group will have prepared a short presentation and will have a diagram of their experimental set up and a table containing their results on the whiteboard. After each group presents their results I will ask them several probing questions to gauge how well they understand the scientific process as well as the concepts being studied.
“What were some sources of error that you think may have affected your results?”
“How could your procedure be adapted if we were working with faster projectiles?”
“What happened if you didn’t shoot the projectile perfectly horizontal?”
“Are these results something that you would have expected?”
After all of the groups have presented their findings to the class I will summarize the findings and reiterate the “bullet shot/dropped” theory. I will explain the implications of this idea and how it can be used to help us study projectiles. This theory implies that the x and the y directions are entirely independent of one another and can be mathematically solved separately. The only variable that remains the same for the two directions is the independent variable of time.
I will instruct the students to take a seat and I will have them all open their notebooks so that these concepts can be formally presented to them.
The notes that I will give the students can be seen on the pages that follow this lesson plan. I will begin by explaining what a projectile is and what “projectile motion” actually means. I will then review the equations that they used in 1 dimensional kinematics and will explain how these can be used for the x and y directions separately.
“Why do we have two sets of equations for the x and y directions?”
We will then discuss how these equations can be simplified for an object that is launched into the air.
“What is the acceleration in the y direction for an object that is launched into the air?”
“What is the acceleration in the x direction for an object that is launched into the air? Why? What are the implications of this?”
I will then show them the simplified versions of the equations and we will proceed to review breaking components into vectors and finally we will do two examples together as a class. Prior to the examples however, I will distribute the graphic organizer that is attached and we will discuss/review the steps to problem solving because this is a critical skill that they must all master.
“Does anyone remember the steps to problem solving? I have made a graphic organizer to help you remember them!”
EXPLORE A:
After distributing the graphic organizer and working through the two example problems together as a class, I will ask the students if they have any specific questions regarding the topics that we covered. I will then have the students work in groups to complete several problems on their own. I will inform them that I will be collected them at the end of the class and they will receive a completion grade for their work. I will require that they explicitly write out all of the steps that they used during the process. By this I mean that I will be checking to ensure that they explicitly identify what variable/variables are given in the problem and draw a diagram to go along with the given scenario. I will also be checking to make sure that they identify the appropriate equations and keep the variables in the equations until the very end (i.e make symbolic algebraic manipulations before plugging in numbers).
Note: Each student will have to turn in their own work.
If groups are having trouble with the task I will offer some guidance:
“What are we trying to solve for in this problem?”
“What would the diagram look like to go along with this situation?”
“How can we figure out which equation will help us in this situation?”
“Given the initial velocity and the angle that the projectile is launched at, what two variables can we determine using simple trigonometric ratios?”
I will essentially guide the groups in the right direction with the problems but I will not explicitly do them for them. I will help them follow the problem solving steps and try to get them in the habit of always resorting back to those basic steps.
If certain groups are finding the problems very simple, I will have more challenging problems to present to them.
The problems that I will give to the students can be seen in the attached document and were found in “Barrons Guide to AP Physics”
SUMMARIZE/SHARE A:
Once all of the groups have finished “exploring” the sample problems we will reconvene as a class and I will ask them if they have any specific questions about the problems. If they do then we will address those and possibly do the problems together as a class.
If there are no questions and there is still time remaining I will give the students a HW assignment from their textbook “Holt Physics”
I will also explain to the students that they will have a short quiz at the beginning of the following period on basic projectile motion problems.
HW: Pg 97 #1-4
Launch A:
During the first portion of this lesson I will be able to observe the students as they present their findings from the investigation that was performed on the previous day. The students will be forced to vocalize their understanding of what they did and what they were able to show. In the process of doing this I will be informally assessing them on the presentation skills and on their knowledge of the content of the lesson. I will specifically be looked to see if they truly grasp the “bullet shot/ dropped” phenomenon. In this process I will also be able to assess whether or not the students work well in groups. I will take notes during each presentation and will determine if it will be necessary to spend more class time on the concepts that were taught the previous day. If I deem it necessary I will postpone the next instructional period and will elaborate on the previous concepts. It is very important to build a conceptual foundation before introducing mathematics into the physics classroom and I will use informal assessment to make sure that this happens.
Explore/ Summarize A:
During the group work I will walk around the room and assist the students. In the process of doing this I will be informally assessing the students on their understanding of the material. If some students seem to be struggling I will be sure to spend more time with their groups and attempt to clear up any misconceptions that they may have and fill in any gaps in their knowledge that may exist. I will also collect the HW and will formally assess each students ability to follow the problem solving process. I will not assess whether or not they got the problem correct (although I will check this as well) but rather I will make sure that they followed the correct procedure and identified the steps properly. I want to stress problem solving and critical thinking skills and I will constantly make sure that the students are showing all of their work and are breaking down problems effectively.
This section to be completed only if lesson plan is implemented.